Place the bed sheet in a freezer for an hour to allow the gum to freeze. When the gum is frozen solid, remove the sheet from the freezer and slide the backside of a knife underneath it to try to lift it. The hard gum will often lift in one piece, making cleanup easier than when the gum is soft and malleable.
Spray the gum with a water-displacing lubricant such as WD-40 if you don't have a freezer or using the freezer is not convenient. Allow the spray to soak in to the bed sheets for a couple minutes.
Rub the gum and spray with a rag to lift the gum away from the bed sheet. Water-displacing sprays work effectively as a stain and sticky substance remover. With a few scrubs, you will lift the gum off the sheets. After removing the gum, run the sheets through the washing machine to remove the residue left by the spray.